Copulas with discrete and mixed marginals are very increasingly popular [1]}, although parameter estimation is challenging.
Several approximate likelihood
approaches based on the continuous extension of discrete random variables studied by [2]} have been suggested, although these typically exhibit significant bias in the copula parameter estimates;
see [3]} and [4]} for demonstrations using the Gaussian copula.
In contrast, Bayesian data augmentation approaches discussed here can evaluate
the posterior of such parametric copula models in high dimensions without
resorting to approximating the likelihood.
For implicit copulas, estimation using data augmentation based on the extended likelihood
in Section REF  is popular in practice due to its simplicity and robustness. When using MCMC
sampling, as in [5]} for the Gaussian copula, the posterior is evaluated exactly (up to Monte Carlo error) and can be
used in high dimensions as demonstrated by [6]} and [7]}.
An alternative approach that can be used in even higher dimensions is variational inference,
as [8]} outline, although this is an approximate estimation method.
